Yes, we now know that there is no explanation that Einstein would have been satisfied with.

This is Bell's theorem - there is no local realistic explanation for quantum mechanics.

More generally, there is at least one explanation of the measurement problem for non-relativistic quantum mechanics - this explanation is called Bohmian Mechanics. Einstein knew about Bohmian Mechanics, but he didn't like it, because Bell's theorem was not discovered until after Einstein died.

I don't know whether Einstein would have been satisfied with it, but there is a resolution of the measurement problem that preserves locality. It's called consistent histories. It's just a version of Copenhagen, which interprets time evolution as a stochastic process in the framework of quantum mechanics. The similarities to the theory of classical stochastic processes are striking, so it seems like a very natural way to explain quantum mechanics and. The resolution of the measurement problem and locality follows in a natural way, so Einstein would certainly at least have considered it a worthwhile candidate after he had learned about Bell's theorem.

I don't know whether Einstein would have been satisfied with it, but there is a resolution of the measurement problem that preserves locality. It's called consistent histories. It's just a version of Copenhagen, which interprets time evolution as a stochastic process in the framework of quantum mechanics. The similarities to the theory of classical stochastic processes are striking, so it seems like a very natural way to explain quantum mechanics and. The resolution of the measurement problem and locality follows in a natural way, so Einstein would certainly at least have considered it a worthwhile candidate after he had learned about Bell's theorem.

But of course, it is not realistic, unlike classical stochastic processes.

But of course, it is not realistic, unlike classical stochastic processes.

Yes, like I said, it's just a version of Copenhagen. But it explains nicely the appearance of the projection operators in ##\mathrm Tr(P_n U(t_n) \cdots P_1 U(t_1)\rho)##, since the probabilities in classical stochastic processes are defined using the exact same formula with the only restriction being that only commuting ##P_n## are allowed. So it becomes apparent that the projections don't contribute to time evolution and observation plays no distinguished role. In that sense, quantum time evolution is just a non-commutative generalization of a classical stochastic process.

Yes, like I said, it's just a version of Copenhagen. But it explains nicely the appearance of the projection operators in ##\mathrm Tr(P_n U(t_n) \cdots P_1 U(t_1)\rho)##, since the probabilities in classical stochastic processes are defined using the exact same formula with the only restriction being that only commuting ##P_n## are allowed. So it becomes apparent that the projections don't contribute to time evolution and observation plays no distinguished role. In that sense, quantum time evolution is just a non-commutative generalization of a classical stochastic process.

It is not agreed on that observation plays no distinguished role in consistent histories - who chooses which consistent family occurs?
For example, https://arxiv.org/abs/quant-ph/0209123 comments "In the history interpretation, the paradox transposes in terms of choice of families of histories: the problem is that there is no way to eliminate the families of histories where the cat is at the same time dead and alive; actually, most families that are mathematically acceptable through the consistency condition contain projectors on macroscopic superpositions, and nevertheless have exactly the same status as th families that do not. One would much prefer to have a “super-consistency” rule that would eliminate these superpositions; this would really solve the problem, but such a rule does not exist for the moment. At this stage, one can then do two things: either consider that the choice of sensible histories and reasonable points of view is a matter of good sense - a case in which and one returns to the usual situation in the traditional interpretation, where the application of the postulate of wave packet is also left to the good taste of the physicist; or invoke decoherence and coupling to the external world in order to eliminate all these unwanted families - a case in which one returns to the usual situation where, conceptually, it is impossible to ascribe reasonable physical properties to a closed system without refereeing to the external world and interactions with it, which opens again the door to the Wigner friend paradox, etc."

There is also implicit criticism by https://arxiv.org/abs/1106.0767 which attempts to fix a problem with consistent histories - or do you think they are attempting to fix a non-existent problem?

It is not agreed on that observation plays no distinguished role in consistent histories - who chooses which consistent family occurs? Or do you think there is no need for attempts such as https://arxiv.org/abs/1106.0767 to fix a non-existent problem?

That's not related to a distinguished role of observers. In consistent histories, you need to choose a set of alternatives that you want to assign probabilities to. However, the physics is independent of this choice. The paper is concerned with the question of whether one choice can be singled out by asking the alternatives to behave classically. It's however not required that this choice must be made.

That's not related to a distinguished role of observers. In consistent histories, you need to choose a set of alternatives that you want to assign probabilities to. However, the physics is independent of this choice. The paper is concerned with the question of whether one choice can be singled out by asking the alternatives to behave classically. It's however not required that this choice must be made.

The physics is not independent of this choice - in particular, predictions are not independent of this choice.

The physics is independent of the choice. There is an analogy in classical probability theory: If you throw a die, you can choose different sets of mutually exclusive events:
One example is {1}, {2,3}, {4,5,6}
Another example is {1,2,3}, {4,5}, {6}
A third example is {1}, {2}, {3}, {4}, {5}, {6}
You can assign probabilities to the choices in the following way:
First example: P({1}) = 1/6, P({2,3}} = 2/6, P({4,5,6}) = 3/6
Second example: P({1,2,3}) = 3/6, P({4,5}) = 2/6, P({6}) = 1/6
Third example: P({i}) = 1/6
In classical probability you can always find a choice such that the other choices arise from this choice by taking unions and the probabilities are compatible. This would be the case for example 3. In consistent histories, it's not the case, that there is a "smallest" choice, because the structure of the event algebra is different than in classical probability. It's nevertheless the case that the probabilities are compatible. The physically predicted probabilities of events that are shared by different sets of mutually exclusive alternatives are equal and if the events are such that you can take their conjunction or disjunction, the probabilities will also be compatible. No physical prediction in one choice will differ from a physical prediction in another choice. It's just the case that it sometimes doesn't make sense to take the conjunction of events. For example the event "spin z = 1/2, spin x = 1/2" is just not an admissible event and hence you can't find a decomposition.

Phrased differently: No matter which set of mutually exclusive alternatives I choose, the experimental results will be compatible with this choice. If two physicists choose to use two different sets, they will both be right, even if they match their sets to the exact same experimental data.

Do we yet have an explanation for the Measurement Problem that Einstein would have been satisfied with?

Nobody knows - since this would require time travel. I don't think there has been one that would fit with the point of view Einstein consistently expressed in the Einstein-Bohr letters, but, as a scientist, Einstein would have to take the modern point of view seriously even if he does not find it satisfying.
Nature does not care how one particular person feels about it.

What do you think he would say when confronted with the results of the quantum eraser experiment?

Nbody can know for sure and it doesn't matter. What matters is how the experiment informs the way physical models work today and into the future.

However, from other posts by this author - I think we should not assume that @pittsburghjoe understands "the measurement problem" and "the quantum erasor experiment" even at the pop-science level. So it is unclear what he is actually asking here.

For the rest - can we agree that there is still robust discussion within the physics community about how to handle this stuff? I think that may be part of why the question was asked in the first place.